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To divide a polynomial by a binomial of the form x - c using synthetic division. Use the Remainder Theorem in conjunction with synthetic division to find a functional value. Use the Factor Theorem in conjunction with synthetic division to find factors and zeros of a polynomial function.

Synthetic division is another way to divide a polynomial by the factor  [tex](x - k)[/tex] , where k is a constant.

  • Synthetic division is usually used to find factors of polynomials of degree 3 that can not be factored by grouping.
  • Synthetic division is also used to factor polynomials of degree 4 or higher.
  • We use synthetic division if no other methods will work since there is a trial and error aspect to using it.

Example, We have to factorize  [tex]f(x)=x^{3}+3x^{2} -4[/tex]

Substitute x = 1 in above function.

     [tex]f(1)=1^{3}+3-4=0[/tex]

So that, (x - 1) will be a factor of above function.

Now divide given function by (x - 1)

              [tex]\frac{x^{3}+3x^{2} -4}{x-1}=x^{2} +4x+4=(x+2)^{2}[/tex]

Factors of function is,

         [tex]x^{3}+3x^{2} -4=(x-1)(x+2)^{2}[/tex]

This is known as  synthetic division method to find factors.

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